A hybrid forecasting model for enrollments based on aggregated fuzzy time series and particle swarm optimization

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Abstract

In this paper, a new forecasting model based on two computational methods, fuzzy time series and particle swarm optimization, is presented for academic enrollments. Most of fuzzy time series forecasting methods are based on modeling the global nature of the series behavior in the past data. To improve forecasting accuracy of fuzzy time series, the global information of fuzzy logical relationships is aggregated with the local information of latest fuzzy fluctuation to find the forecasting value in fuzzy time series. After that, a new forecasting model based on fuzzy time series and particle swarm optimization is developed to adjust the lengths of intervals in the universe of discourse. From the empirical study of forecasting enrollments of students of the University of Alabama, the experimental results show that the proposed model gets lower forecasting errors than those of other existing models including both training and testing phases.

Research highlights

► A new forecasting model based on fuzzy time series and particle swarm optimization is developed. ► The lengths of intervals are adjustable in the universe of discourse. ► The proposed model gets lower forecasting errors than those of other existing models.

Introduction

In the past decades, many forecasting models have been developed to deal with various domain problems to help people to make decisions, such as crop production (Singh, 2007a, Singh, 2007b), academic enrollments (Chen, 1996, Chen and Chung, 2006a, Chen and Chung, 2006b, Esogbue and Qiang, 1998, Li and Cheng, 2007, Song and Chissom, 1993b, Song and Chissom, 1994), stock markets (Cheng et al., 2008, Chu et al., 2009, Teoh et al., 2009, Wang and Chen, 2009) and temperature prediction (Lee et al., 2007, Wang and Chen, 2009). The tradition forecasting methods cannot deal with forecasting problems in which the historical data needs to be represented by linguistic values. Fuzzy set theory was firstly presented by Zadeh (1965) to handle problems with linguistic values. The concepts of fuzzy sets (Zadeh, 1978) have been successfully adopted to time series by Song and Chissom (1993). They developed two forecasting models, the time-invariant fuzzy time series model and the time-variant fuzzy time series model (Song and Chissom, 1993b, Song and Chissom, 1994), to forecast the students in the enrollments of the University of Alabama. Unfortunately, their method needs max–min composition operations to deal with fuzzy rules. It takes a lot of computation time when fuzzy rule matrix is big. Chen (1996) proposed an efficient first-order fuzzy time series model which consists of simple arithmetic calculations only. After that, fuzzy time series has been widely studied for improving accuracy of forecasting in many applications.

Huarng (2001a) presented effective approaches which can properly adjust the lengths of intervals to get better forecasting accuracy. Huarng (2001b) also presented heuristic models to improve forecasting accuracy of fuzzy time series. Hwang, Chen, and Lee (1998) proposed a time-variant fuzzy time series model using the variation of historical data between current year and past years on enrollments. Chen (2002) proposed a new forecast model based on the high-order fuzzy time series to forecast the enrollments of University of Alabama. Yu (2005) presented a new model which can refine the lengths of intervals during the formulation of fuzzy relationships and hence capture the fuzzy relationships more appropriately. Both the stock index and enrollment are used as the targets in the empirical analysis. Chen and Chung, 2006a, Chen and Chung, 2006b presented the first-order and high-order fuzzy time series model to deal with forecasting problems based on genetic algorithms. Singh, 2007a, Singh, 2007b presented simplified and robust computational methods for the forecasting rules based on one and various parameters as fuzzy relationships, respectively. Li and Cheng (2007) proposed a novel deterministic forecasting approach to effectively partitioning intervals and achieved forecasting accuracy with different interval lengths. Singh (2008) proposed a fuzzy time series model of order three and used a time variant difference parameter on current state to forecast the next state. Chen, Cheng, and Teoh (2008) presented a high order fuzzy time series model derived from the multi-period adaptation model and the adaptive expectation model. Kuo et al. (2009a) presented a new hybrid forecasting model which combined particle swarm optimization with fuzzy time series to find proper length of each interval.

From the literature listed above, adapting lengths of intervals and creating forecasting rules are two important issues considered to be critical influencing the forecasting accuracy. Especially, Huarng pointed out that effective length of intervals can improve forecasting accuracy in fuzzy time series model. To reconcile this issue, Huarng, 2001a, Huarng, 2001b introduced two methods to adjusted interval lengths on Chen’s fuzzy time series model. One method applied two different lengths, the average-based length and the distribution-based length, respectively. Another method modified previous method by using the ratio-based length to get better forecasting accuracy. Recently, Kuo et al. (2009a) formulated finding proper lengths of intervals to optimization problem and applied particle swarm optimization approach on Chen’s model (Chen, 1996). As known the performance of optimization problem is determined by objective functions. Chen defined the forecasting values based on defuzzified historical data in fuzzy time series model. After exploring the experimental results of corresponding literatures, we observe the forecasting accuracy is impacted not only by the fuzzy logical relationships but also by the latest historical data. Therefore, it may reduce the forecasting accuracy when the variation of latest time cannot be accounted into forecasting factors. To reconcile this problem, first a new method which aggregates global information of fuzzy logical relationships with local information of latest fuzzy fluctuation (LFF) is developed to find forecasting values. Then, the mean square error (MSE) value is applied to estimate the forecasting accuracy. Finally, a new hybrid forecasting model, called AFPSO, based on aggregated fuzzy time series and particle swarm optimization (PSO) is developed to adjust the length of each interval in the universe of discourse by minimizing MSE value. The empirical study on the enrollment data at the University of Alabama shows that the performance of our new model is better than those of any existing methods. For both training and testing phases, it produces higher forecasting accuracy on fuzzy time series with various orders and different intervals.

The rest of this paper is organized as follows. In Section 2, a brief review of the schemes of fuzzy time series and particle swarm optimization are introduced. Section 3, first gives the details of fuzzy time series model based on global and local information, and then demonstrates how to use the particle swam optimization algorithm to find the effective lengths of intervals in the universe of discourse during training phase. Section 4 evaluates the forecasting performance of the proposed method with the existing methods on the data set of enrollments of students of the University of Alabama. Finally, some concluding remarks are discussed in Section 5.

Section snippets

Literature review

In this section, we briefly review the basic concepts of fuzzy time series and particle swarm optimization.

Fuzzy time series

In this section, we introduce our method to forecast the enrollments of the University of Alabama. The historical data of enrollments of the University of Alabama are listed in Table 1. The step-wise procedure of the proposed model of fuzzy time series is detailed as follows:

Step 1: Define the universe of discourse

Assume Y(t) be the historical data of enrollments at year t (1971  t  1993). The university of discourse is defined as U = [starting, ending], where starting  Y(t)  ending. According to the

Experimental results

To illustrate the forecasting performance of the proposed method, the actual enrollments of students of the University of Alabama are used as data set in both training and testing phases. The data set of enrollments is covered from year 1971 to 1992. The experimental results of AFPSO model are compared with those of corresponding models for various order and different intervals.

Conclusions

In this paper, we have presented a hybrid forecasting model in academic enrollments based on two advanced methods, fuzzy time series and particle swarm optimization. In order to improve the forecasting accuracy of the HPSO model (Kuo et al., 2009a), we aggregate the global information of fuzzy relationships with the local information of latest fuzzy fluctuation to find the defuzzified forecasting value. Then, a novel hybrid forecasting model based on aggregated fuzzy time series and particle

Acknowledgements

This work was supported in part by the National Science Council under contract number NSC-98-2219-E-011-002, NSC-98-2221-E-011-133-MY3, NSC-98-2923-E-011-004-MY3, NSC-99-2916-I-011-002-A1, and it was also partially supported by the 111 Project under the Grant No. 111-2-14.

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